A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

The three-boson system with short-range interactions

We discuss renormalization of the non-relativistic three-body problem with short-range forces. The problem is non-perturbative at momenta of the order of the inverse of the two-body scattering length. An infinite number of graphs must be summed, which leads to a cutoff dependence that does not appear in any order in perturbation theory. We argue that this cutoff dependence can be absorbed in one local three-body force counterterm and compute the running of the three-body force with the cutoff. This allows a calculation of the scattering of a particle and the two-particle bound state if the corresponding scattering length is used as input. We also obtain a model-independent relation between binding energy of a shallow three-body bound state and this scattering length. We comment on the power counting that organizes higher-order corrections and on relevance of this result for the effective field theory program in nuclear and molecular physics.     

Three-body recombination in bose gases with large scattering length

An effective field theory for the three-body system with large scattering length is applied to three-body recombination to a weakly bound s-wave state in a Bose gas. Our model independent analysis demonstrates that the three-body recombination constant alpha is not universal, but can take any value between zero and 67.9Planck's over 2pia(4)/m, where a is the scattering length. Other low-energy three-body observables can be predicted in terms of a and alpha. Near a Feshbach resonance, alpha should oscillate between those limits as the magnetic field B approaches the point where a-->infinity. In any interval of B over which a increases by a factor of 22.7, alpha should have a zero.     

Three-boson problem near a narrow Feshbach resonance

We consider a three-boson system with resonant binary interactions and show that for sufficiently narrow resonances three-body observables depend only on the resonance width and the scattering length. The effect of narrow resonances is qualitatively different from that of wide resonances revealing novel physics of three-body collisions. We calculate the rate of three-body recombination to a weakly bound level and the atom-dimer scattering length and discuss implications for experiments on Bose-Einstein condensates and atom-molecule mixtures near Feshbach resonances.     

Borromean three-body heteroatomic resonances

We present an approach to analyze recent experimental evidences of Efimov resonant states in mixtures of ultracold gases, by considering two-species three-body atomic systems bound in a Borromean configuration, where all the two-body interactions are unbound. For such Borromean three-body systems, it is shown that a continuum three-body s-wave resonance emerges from an Efimov state as a scattering length or a three-body scale is moved. The energy and width of the resonant state are determined from a scaling function with arguments given by dimension-less energy ratios relating the two-body virtual state subsystem energies with the shallowest three-body bound state. The peculiar behavior of such resonances is that their peaks are expected to move to lower values of the scattering length, with increasing width, as one raises the temperature. For Borromean systems, two resonant peaks are expected in ultralow-temperature regimes, which will disappear at higher energies. It is shown how a Borromean-Efimov excited bound state turns out to a resonant state by tuning the virtual two-body subsystem energies or scattering lengths, with all energies written in units of the next deeper shallowest Efimov state energy. The resonance position and width for the decay into the continuum are obtained as universal scaling functions (limit cycle) of the dimensionless ratios of the two and three-body scales, which are calculated numerically within a zero-range renormalized three-body model.     

Stability and Chaos of Two Coupled Bose-Einstein Condensates with Three-Body Interaction

We study the dynamics of two Bose-Einstein condensates (BECs) tunnel-coupled by a double-well potential.A real three-body interaction term is considered and a two-mode approximation is used to derive two coupled equations,which describe the relative population and relative phase. By solving the equations and analyzing the stability of the system, we find the stable stationary solutions for a constant atomic scattering length. When a periodically time-varying scattering length is applied, Melnikov analysis and numerical calculation demonstrate the existence of chaotic behavior and the dependence of chaos on the three-body interaction parameters.     

Realization of a resonant Fermi gas with a large effective range

We have measured the interaction energy and three-body recombination rate for a two-component Fermi gas near a narrow Feshbach resonance and found both to be strongly energy dependent. Even for de Broglie wavelengths greatly exceeding the van der Waals length scale, the behavior of the interaction energy as a function of temperature cannot be described by atoms interacting via a contact potential. Rather, energy-dependent corrections beyond the scattering length approximation are required, indicating a resonance with an anomalously large effective range. For fields where the molecular state is above threshold, the rate of three-body recombination is enhanced by a sharp, two-body resonance arising from the closed-channel molecular state which can be magnetically tuned through the continuum. This narrow resonance can be used to study strongly correlated Fermi gases that simultaneously have a sizable effective range and a large scattering length.     

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent.The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles(bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length,phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

Analytical Solution of Partial-Wave Faddeev Equations with Application to Scattering and Statistical Mechanical Properties

In this work, the Faddeev equations for three-body scattering at arbitrary angular momentum are exactly solved and the transition matrices for some transition processes, including scattering and rearrangement channels are formulated in terms of free-particle resolvent matrix. A generalized Yamaguchi rank-two nonlocal separable potential has been used to obtain the analytical expressions for partial wave scattering properties of a three-particle system. The partial-wave analysis for some transition processes in a three-particle system is suggested. The partial-wave three-particle transition matrix elements have been constructed via knowledge of the matrix elements of the free motion resolvent. The calculation of a number of scattering properties of interest of the system such as transition matrix and its poles (bound states and resonances) and consequently other related quantities like scattering amplitudes, scattering length, phase shifts and cross sections are feasible in a straightforward manner. Moreover, we obtain a new analytical expression for the third virial coefficient in terms of three-body transition matrix.     

The system of differential equations in the momentum space for the three-body problem: Continuous spectrum

A system of differential equations in the momentum space for describing three-body scattering is obtained. Preliminary calculations of the length of boson scattering on a bound bosonic pair are performed. The data obtained are in good agreement with the results of calculations of helium atom scattering on a helium dimer.     

Three-Body System with Two-Channel Zero-Range Interaction Model of Feshbach Resonance

We perform three-body calculations of trimers and atom-dimer scattering near a Feshbach resonance using two interaction models. The first model is a one-channel zero-range model, where the scattering length follows the phenomenological dependence on the external magnetic field. The second is a two-channel model capable to describe the Feshbach resonance. The scattering length dependence on magnetic detuning is recovered. We compare the predictions of these two models, and show that near a Feshbach resonance important differences are expected.     

Binding and pauli principle corrections in subthreshold pion-nucleus scattering

In this investigation I develop a three-body model for the single scattering optical potential in which the nucleon binding and the Pauli principle are accounted for. A unitarity pole approximation is used for the nucleon-core interaction. Calculations are presented for the π−⁴He elastic scattering cross sections at energies below the inelastic threshold and for the real part of the π−⁴He scattering length by solving the three-body equations. Off-shell kinematics and the Pauli principle are carefully taken into account.

The binding correction and the Pauli principle correction each have an important effect on the differential cross sections and the scattering length. However, large cancellations occur between these two effects. I find an increase in the π−⁴He scattering length by 100%; an increase in the cross sections by 20–30% and shift of the minimum in π−⁴He scattering to forward angles by 10°.     

Three-Body Recombination with Two-Channel Contact Interactions

We use a two-channel contact interaction model to describe a system of three identical bosons. The two-channel model quantitatively describes the phenomena of Feshbach resonance in agreement with the phenomenological expression relating scattering length to magnetic detuning. The model also has a finite effective range. We investigate finite range effects in three-body recombination. The simpler one-channel contact interaction model predicts a characteristic geometric scaling of minima in the recombination coefficient as a function of scattering length with scaling parameter 22.7. We show that this factor is reduced when the effective range is included. We compare calculations to experiment.     

Optimum spin squeezing in Bose-Einstein condensates with particle losses

The problem of spin squeezing with a bimodal condensate in the presence of particle losses is solved analytically by the Monte Carlo wave function method. We find the largest obtainable spin squeezing as a function of the one-body loss rate, the two-body and three-body rate constants, and the s-wave scattering length.     

Fluctuation assisted collapses of Bose–Einstein condensates

We study the collapse dynamics of a Bose–Einstein condensate subjected to a sudden change of the scattering length to a negative value by adopting the self-consistent Gaussian state theory for mixed states. Compared to the Gross–Pitaevskii and the Hartree–Fock–Bogoliubov approaches, both fluctuations and three-body loss are properly treated in our theory. We find a new type of collapse assisted by fluctuations that amplify the attractive interaction between atoms. Moreover, the calculation of the fluctuated atoms, the entropy, and the second-order correlation function showed that the collapsed gas significantly deviated from a pure state.     

Scattering length scaling laws for ultracold three-body collisions

We present a simple and unifying picture that provides the energy and scattering length dependence for all inelastic three-body collision rates in the ultracold regime for three-body systems with short-range two-body interactions. Here, we present the scaling laws for vibrational relaxation, three-body recombination, and collision-induced dissociation for systems that support s-wave two-body collisions. These systems include three identical bosons, two identical bosons, and two identical fermions. Our approach reproduces all previous results, predicts several others, and gives the general form of the scaling laws in all cases.     

Ground-state properties of dilute one-dimensional Bose gas with three-body repulsion

We determined perturbatively the low-energy universal thermodynamics of dilute one-dimensional bosons with the three-body repulsive forces. The final results are presented for the limit of vanishing potential range in terms of three-particle scattering length. An analogue of Tan's energy theorem for considered system is derived in generic case without assuming weakness of the interparticle interaction. We also obtained an exact identity relating the three-body contact to the energy density.     

Atom-dimer scattering for confined ultracold fermion gases

We solve the three-body problem of a quasi-one-dimensional ultracold Fermi gas with parabolic confinement length a (perpendicular) and 3D scattering length a. On the two-body level, there is a Feshbach-type resonance at a (perpendicular)/a approximately 1.46, and a dimer state for arbitrary a (perpendicular)/a. The three-body problem is shown to be universal, and described by the atom-dimer scattering length a(ad) and a range parameter b(ad). In the dimer limit a (perpendicular)/a>1, we find a repulsive zero-range atom-dimer interaction. For a (perpendicular)/a<-1, however, the potential has long range, with a(ad)>0 and b(ad)>a(ad). There is no trimer state, and despite a(ad)=0 at a( perpendicular)/a approximately 2.6, there is no resonance enhancement of the interaction.     

General three-particle scattering theory

A unified treatment of three-particle scattering theory with a three-body force in addition to the usual pair interactions is developed. The relationship of the generalized AGS and Faddeev formalisms to each other as well as distinct versions of each corresponding to the two most natural techniques for handling the three-body potential are established. It is found, just as in the case without the three-particle force, that the AGS formalism appears to be more practical for considering elastic and rearrangement scattering in two-body channels. On the other hand, for scattering amplitudes with at least one three-body channel (breakup and the 3-to-3) the Faddeev version of the theory is preferable. Other advantages of each formalism depending upon the treatment of the three-body interaction are noted.     

Operation of the Faddeev Kernel in Configuration Space

We present a practical method to solve Faddeev three-body equations at energies above the three-body breakup threshold as integral equations in coordinate space. This is an extension of a previously used method for bound states and scattering states below three-body breakup threshold energy. We show that breakup components in three-body reactions produce long-range effects on Faddeev integral kernels in coordinate space, and propose numerical procedures to treat these effects. Using these techniques, we solve Faddeev equations for neutron-deuteron scattering to compare with benchmark solutions.